An Optimal Policy with Three-Parameter Weibull Distribution Deterioration, Quadratic Demand, and Salvage Value Under Partial Backlogging

An Optimal Policy with Three-Parameter Weibull Distribution Deterioration, Quadratic Demand, and Salvage Value Under Partial Backlogging

Trailokyanath Singh (C. V. Raman College of Engineering, Department of Mathematics, Odisha, India), Hadibandhu Pattanayak (Institute of Mathematics and Applications, Department of Mathematics, Odisha, India), Ameeya Kumar Nayak (IIT Roorkee, Department of Mathematics, Roorkee, India) and Nirakar Niranjan Sethy (Ravenshaw University, Odisha, India)
Copyright: © 2018 |Pages: 20
DOI: 10.4018/IJRSDA.2018010106
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This paper deals with an EOQ (Economic Order Quantity) model for deteriorating items having the following characteristics: 1) Deteriorating items follow a three-parameter Weibull distribution deterioration rate; 2) Shortages are allowed and are partially backlogged; 3) Salvage value of items is incorporated; 4) Demand is deterministic and a time-dependent quadratic function of time. The principal objective of the introduced model is to minimize the average total inventory cost by finding an optimal replenishment policy. The effectiveness of the model is validated with a numerical example and the sensitivity analysis of the optimal solutions to changes in the values of the various parameters associated with the model has been performed.
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Traditionally, one of the assumptions of inventory models was that the items preserve their original characteristics throughout while they are kept stored in the inventory. This assumption is not valid for all items. However, physical goods like vegetables, grains, blood, meat, pharmaceuticals, fashion goods, volatile or radioactive substances etc. deteriorate over time. Deterioration is defined as the damage or decay, spoilage, vaporization, dryness etc. Deterioration arises due to some physical changes in the items which make the item value dull. Thus, the control and maintenance of inventories of deteriorating items is a big challenge in any modern organization. In the classical inventory analysis, the mathematical modeling of inventory control started with the work of Harris (1913) who assumed implicitly that the stocked items have infinite shelf lives. Later, Wilson (1934) generalized the EOQ model of Harris’. Whitin (1957), the first researcher, accounted for the deterioration of fashion goods at the end of their prescribed storage period. Ghare and Schrader (1963), were the first researchers to extend the classical EOQ formula by including the negative exponential decaying inventory. Shah and Jaiswal (1977) studied an order level inventory model for deteriorating items assuming a constant deterioration rate. Later, Aggarwal (1978) modified the work of Shah and Jaiswal by calculating the average inventory cost.

Another class of inventory models has been developed along the time-dependent deterioration rate. Covert and Philip (1973) studied the inventory model by using the two-parameter Weibull distribution deterioration with the distribution of time bound deterioration. Later, Philip (1974) extended Covert and Philip’s using three-parameter Weibull distribution of time to deterioration. Misra (1975) analyzed the inventory model with some special assumptions using two-parameter Weibull deterioration rate, finite replenishment rate and without shortage. However, for all these inventory systems, the demand rate is considered as constant. Later, several researchers developed papers by analyzing the deteriorating inventory, such as Dave and Patel (1981), Hariga and Benkherouf (1994) and others. Jaggi et al. (2013) presented a two-warehouse inventory problem with constant demand rate and shortages under inflationary conditions. Aggarwal and Tyagi (2014) studied the optimal policies for deteriorating items with credit linked demand under two levels of trade credit financing in the inventory system. A detailed study regarding the inventory model for deteriorating items was given in the review articles of Nahmias (1982), Raafat (1991), Goyal and Giri (2001), Li et al. (2010).

In real life situations, customers demand sometimes cannot be fulfilled by the suppliers from the existing stocks. This situation is known as a shortage condition or stock-out situation. Harris (1913) did not include the shortage condition in his model. Researches were carried out by assuming shortage period in the demand either as backlogged or lost. But in general, it is seen some customers are willing to wait for the next replenishment. In case of complete backlogging, customers are willing to wait and the supplier is committed to meet all the demands with no lost sale. But in case of partial backlogging only a fraction of the missed demand is allowed to be fulfilled from the replenished stock. The length of waiting time for the next replenishment is the main factor in calculating the backlogging rate. The longer the waiting time is, the smaller the backlogging rate would be and vice versa.

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